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Interactions between Polymers andNanoparticles : Formation of

« Supermicellar » Hybrid AggregatesJ.-F. Berret@, K. Yokota* and M. Morvan,

Complex Fluids Laboratory, UMR CNRS - Rhodia n°166,Cranbury Research Center Rhodia 259 Prospect Plains Road

Cranbury NJ 08512 USA

Abstract :

When polyelectrolyte-neutral block copolymers are mixed in solutions to oppositely

charged species (e.g. surfactant micelles, macromolecules, proteins etc…), there is the

formation of stable “supermicellar” aggregates combining both components. The resulting

colloidal complexes exhibit a core-shell structure and the mechanism yielding to their

formation is electrostatic self-assembly. In this contribution, we report on the structural

properties of “supermicellar” aggregates made from yttrium-based inorganic nanoparticles

(radius 2 nm) and polyelectrolyte-neutral block copolymers in aqueous solutions. The

yttrium hydroxyacetate particles were chosen as a model system for inorganic colloids, and

also for their use in industrial applications as precursors for ceramic and opto-electronic

materials. The copolymers placed under scrutiny are the water soluble and asymmetric

poly(sodium acrylate)-b-poly(acrylamide) diblocks. Using static and dynamical light

scattering experiments, we demonstrate the analogy between surfactant micelles and

nanoparticles in the complexation phenomenon with oppositely charged polymers. We also

determine the sizes and the aggregation numbers of the hybrid organic-inorganic

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complexes. Several additional properties are discussed, such as the remarkable stability of

the hybrid aggregates and the dependence of their sizes on the mixing conditions.

Submitted to Colloids and Surfaces A, 30 October 2004

Writing started on 8 October 2004

Paper presented at the ECIS conference held in Almeria, Sept. 2004.

@ jeanfrancois.berret@us.rhodia.com

* Present address : Rhodia, Centre de Recherches d’Aubervilliers, 52 rue de la Haie Coq,

F-93308 Aubervilliers Cedex France

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I - Introduction

In the phosphor, superconductor and ceramic manufacturing, rare earth oxides and yttrium

oxide in particular are of great importance, and thus they have found multiple industrial

applications [1,2]. In recent years, particles made from these oxides and with sizes down to

the nanometer range (1 – 100 nm) have attracted lots of interest. The use of ultrafine

particles not only favors a good dispersibility or a very highly densification of the final

products [3] but also improve their performances. In luminescent displays for instance,

ultrafine particles are a crucial ingredient because they allow the design of high resolution

panels [4,5]. For applications, the traditional process to obtain fine rare earth oxide

particles (or precursors) is the precipitation of rare earth salts by alkaline solutions,

followed by the separation of the aggregates, the drying and calcination, eventually

accompanied by mechanical milling. With this method, particles remain large, at a few

hundreds nanometers in size. Moreover, their size distribution is not properly controlled. A

number of papers have proposed alternative methods [6,7], using mostly organic solvents

such as in the sol-gel or in the microemulsion processing. In industrial plants however, due

to intrinsic economical and environmental charges associated to the use of organic

solvents, strict limitations are now being applied. The same trend is true for the

manufacturing of the end-products. Therefore, the aqueous processing is a key feature for

the future applications of ultrafine rare earth particles [8,9].

Using a two-step synthesis, we have developed recently ultra-fine yttrium hydroxyacetate

nanoparticles in aqueous solutions that could fulfill the requirements for finding new

materials [10,11]. The yttrium hydroxyacetate particles are spherical in average (radius 2

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nm), rather monodisperse and they are positively charged at neutral pH. They are regarded

as a precursor of yttrium oxide. For colloidal dispersions in general, the issue of the

stability is crucial. The stability of the yttrium precursors at neutral pH is excellent over a

period of several weeks, after which a slow and progressive precipitation is observed. The

origin of this precipitation is not yet understood. In order to improve the stability of the

yttrium hydroxyacetate nanoparticles and then prevent precipitation, we have studied the

complexation of the precursor nanoparticles with oppositely charged block copolymers.

These copolymers placed under scrutiny are the water soluble poly(sodium acrylate)-b-

poly(acrylamide) diblocks, noted in the sequel of the paper PANa-b-PAM. In PANa-b-

PAM, the poly(sodium acrylate) block is electrostatically charged and of opposite charge

to that of the particles and the poly(acrylamide) is neutral. When polyelectrolyte-neutral

block copolymers are mixed in solutions to oppositely charged species (e.g. surfactant

micelles, macromolecules, proteins etc…), there is the formation of stable “supermicellar”

aggregates combining both components [12-30]. The resulting colloidal complexes exhibit

a core-shell structure and the mechanism yielding to their formation is known as

electrostatic self-assembly [14,31]. A schematic representation of the colloidal complexes

made from diblocks and surfactants [22-24,28,29] is displayed in Fig. 1. In this work,

using dynamical and static light scattering, we demonstrate the formation of colloidal

complexes resulting from the spontaneous association of inorganic ultra-fine yttrium

particles and block copolymers. The complexation technique enables to control the

aggregation of the organic-inorganic hybrid colloids in the range 20 - 50 nm.

II - Experimental

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II. 1 - Characterization and Sample Preparation

II.1.1 - Polyelectrolyte-Neutral Diblock Copolymer

In this work, surfactant micelles and yttrium-based nanoparticles have been complexed

with the oppositely charged poly(acrylic acid)-b-poly(acrylamide) block copolymers (Fig.

2). The synthesis of this polymer is achieved by controlled radical polymerization in

solution [32,33]. Poly(acrylic acid) is a weak polyelectrolyte and its ionization degree (i.e.

its charge) depends on the pH. All the experiments were conducted at neutral pH (pH 7)

where ~ 70 % of monomers are negatively charged. In order to investigate the role of the

chain lengths on the formation of complexes, copolymers with different molecular weights

were synthesized [29]. In the present study, we focus on a unique diblock, noted

PANa(69)-b-PAM(840). The two numbers in parenthesis are the degrees of polymerization

of each block. These numbers correspond to molecular weights 5000 and 60000 g mol-1,

respectively. The abbreviation PANa stands here for poly(sodium acrylate), which is the

sodium salt of the polyacid. Static and dynamic light scattering experiments were

conducted for the determination of the weight-average molecular weight

€

Mwpol and the

mean hydrodynamic radius

€

RHpol of the chains. In water, PANa(69)-b-PAM(840) is soluble

and its coil configuration is associated to an hydrodynamic radius

€

RHpol = 8 nm. The

weight-average molecular weight is 68 300 ± 2 000 g mol-1, in good agreement with the

values anticipated from the synthesis. The polydispersity index of the polymers was

estimated by size exclusion chromatography at 1.6.

II.1.2 - Inorganic Nanoparticles

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As already mentioned, we are concerned here with the formation of complexes between

PANa(69)-b-PAM(840) copolymers and yttrium nanoparticles. In order to establish a link

with previous studies, we will recall some data obtained on oppositely charged surfactants

such as the dodecyltrimethylammonium bromide (DTAB) [22-24,28,29]. DTAB is a

cationic surfactant with 12 carbon atoms in the aliphatic chain. it was purchased from

Sigma and used without further purification. Its critical micellar concentration is 0.46 wt.

% (15 mmol l-1).

The synthesis of the yttrium hydroxyacetate nanoparticles is based on two chemical

reactions. One is the dissolution of high purity yttrium oxide by acetic acid and the second

is a controlled reprecipitation of yttrium hydroxyacetate obtained on cooling, yielding for

the average composition of the particles Y(OH)1.7(CH3COO)1.3. We have determined by

light scattering the molecular weight (

€

Mwnano = 27 000 g mol-1) and the hydrodynamic

radius of these nanoparticles (

€

RHnano ~ 2 nm). Zeta potential measurements confirm that

they are positively charged (ζ = + 45 mV) and that the suspensions are stabilized by

surface charges. The stability of Y(OH)1.7(CH3COO)1.3 sols at neutral pH is excellent over

a period of several weeks. The radius of gyration of the particles was finally determined by

small-angle neutron and x-ray scattering. It was found respectively at

€

RGnano = 1.82 nm and

1.65 nm, again in good agreement with the dynamical light scattering result.

II.1.3 - Sample Preparation

The supermicellar aggregates were obtained by mixing a surfactant or a nanoparticle

solution to a polymer solution, both prepared at the same concentration c and same pH. For

the surfactant-polymer complexes, the relative amount of each component is monitored by

the charge ratio Z. For the DTAB surfactant and PANa(69)-b-PAM(840) copolymers, Z is

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given by [DTAB]/(69×[PANa(69)-b-PAM(840)]) where the quantities in the square

brackets are the molar concentrations for the surfactant and for the polymer. Z = 1

describes a solution characterized by the same number densities of positive and negative

chargeable ions. Polymer-nanoparticles complexes were prepared using similar protocols

[11]. For this system, the charge ratio could not be used since the average structural charge

borne by the particles is not known. Here, only the effective charge is known [10] and it is

of the order of

€

4RHnano / lB ~ 11, where

€

lB = 0.7 nm is the Bjerrum length in water [34].

We define instead a mixing ratio X which is the volume of yttrium-based sol relative to

that of the polymer. According to the above definitions, the concentrations in nanoparticles

and polymers in the mixed solutions are cpol = c/(1+X) and cnano = Xc/(1+X).

II. 2 - Light Scattering Techniques and Data Analysis

Static and dynamic light scattering were performed on a Brookhaven spectrometer (BI-

9000AT autocorrelator) for measurements of the Rayleigh ratio R (q,c) and of the

collective diffusion constant D(c). A Lexel continuous wave ionized Argon laser was

operated at low incident power (20 mW - 150 mW) and at the wavelength λ = 488 nm. The

wave-vector q is defined as

€

q = 4πnλ

sin(θ / 2) where n is the refractive index of the

solution and θ the scattering angle. Light scattering was used to determine the apparent

molecular weight

€

Mw,app and the radius of the gyration RG of the supermicellar

aggregates. In the regime of weak colloidal interactions, the Rayleigh ratio follows the

classical expression for macromolecules and colloids [35,36] :

€

K cR (q, c) = 1

Mw,app1+

q2RG23

+ 2A2c (1)

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In Eq. 1, K = 4π2n2(dn/dc)2/NAλ4 is the scattering contrast coefficient (NA is the Avogadro

number) and A2 is the second virial coefficient. The refractive index increments dn/dc

were measured on a Chromatix KMX-16 differential refractometer at room temperature.

The values of the refractive index increments for the different solutions are reported in

Table I. With light scattering operating in dynamical mode, we have measured the

collective diffusion coefficient D(c) in the range c = 0.01 wt. % – 1 wt. %. From the value

of D(c) extrapolated at c = 0, the hydrodynamic radius of the colloids was calculated

according to the Stokes-Einstein relation, RH = kBT/6πη0D0, where kB is the Boltzmann

constant, T the temperature (T = 298 K) and η0 the solvent viscosity (η0 = 0.89×10-3 Pa s).

The autocorrelation functions of the scattered light were interpreted using the method of

cumulants. For the supermicellar aggregates, the diffusion coefficients derived by this

technique appeared to be identical above linear term (i.e. the quadratic, cubic and quartic).

III – Results and Discussion

III. 1 - The Surfactant-Nanoparticle Analogy

In Figs. 3 are displayed the scattering properties of mixed solutions of block copolymers

and oppositely charged surfactants [29]. These properties are the Rayleigh ratio R(q,c) at θ

= 90° and the hydrodynamic radius RH. These Properties are shown as function of the

charge ratio Z. The solutions were prepared as described in the experimental section by

mixing a polymer solution at c = 1 wt. % to the surfactant solution at the same

concentration. Hence, each data points in Figs. 3a and 3b corresponds to a distinct solution.

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At low Z (Z < 1), the scattering intensity is independent of Z and it remains at the level of

the pure polymer. The hydrodynamic radii RH are also close to those of single chains. With

increasing Z, there is a critical charge ratio ZC above which the Rayleigh ratio increases

noticeably. R(q,c) then levels off in the range Z = 1 – 10, and decreases at higher Z values.

Dynamical light scattering performed above ZC reveals the presence of a purely diffusive

relaxation mode, associated to Brownian diffusion. For the PANa(69)- b -

PAM(840)/DTAB, the hydrodynamic radius RH ranges between 50 nm and 70 nm. Such

hydrodynamic values are well above those of the individual components, a result that

suggests the formation of mixed aggregates. The microstructure of such aggregates (Fig. 1)

has been discussed at length in previous reports and we refer to them for a quantitative

description [22-24,28,29]. Because this structure is finally very similar to that of polymeric

micelles made from amphiphilic diblocks [14,17,18], the mixed surfactant-polymer

complexes are sometimes referred to as a “supermicelles” or a “supermicellar” aggregates.

We have repeated the same experiments with nanoparticles and polymers. The yttrium

hydroxyacetate sol was mixed with a c = 1 wt. % PANa(69)-b-PAM(840) solution at

different volumic ratios. Figs. 4a and 4b display the R(q,c) measured at θ = 90° and the

hydrodynamic radius RH. The experimental conditions are the same as those of Fig. 3. The

results between surfactant-polymers and nanoparticle-polymer mixtures are qualitatively

similar. Above a critical value of the mixing ratio noted XC (XC ~ 0.1), the scattering

increases and the hydrodynamic radius saturates at values comprised between 30 nm and

40 nm. As for the surfactant system, dynamical light scattering reveals the formation of

rather monodisperse colloids of hydrodynamic sizes much larger than the individual

components. In electrostatic self-assembly, there is classically a mixing ratio, noted XP at

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which all the components present in the solution react and form complexes [14,30,31]. XP

corresponds roughly to a state where the cationic and anionic charges are in equal amounts.

Experimentally, it is the ratio where the number density of complexes is the largest, i.e.

where the scattering intensity as function of X presents a maximum. From the X-

dependence of the Rayleigh ratio (Fig. 4a), we find for the yttrium–diblock system XP =

0.2. Using the molecular weights of the single components (Table I) and the relationship

€

XP = nnanoMwnano / n polMw

pol , the value of XP allows us to find the number of polymers per

particle involved in the complexation. One gets :

€

n pol nnano ~ 2 (2)

In the sequel of the paper, we assume that the hybrid nanoparticle-polymer complexes,

when they form, are at the preferred composition XP.

€

n pol and

€

nnano are the average

aggregation numbers that characterize the mixed colloids.

Above XP, the intensity decreases with increasing X, and an explanation for this behavior

can be found again in the complexation mechanism. At large X, the nanoparticles are in

excess with respect to the polymers, and thus all the polymers are consumed to build the

supermicellar colloids. Since the polymer concentration decreases with X according to cpol

= c/(1+X), the number of these colloids and thus the scattering intensity decreases.

Assuming furthermore

€

n pol to be a constant versus X, the intensity should explicitly

decrease as 1/X. As shown in Fig. 4b by a straight line in the double logarithmic plot, this

behavior is indeed observed for the yttrium-polymer system. At X > 10, we find actually a

state of coexistence between the mixed nanoparticle-polymer aggregates and the

uncomplexed nanoparticles at RH ~ 2 nm. By comparing Figs. 3 and Figs. 4, we

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demonstrate finally the analogy between surfactant micelles and nanoparticles in the

complexation phenomenon with oppositely charged polymers.

III.2 – Hydrodynamic Sizes for the nanoparticle-polymer complexes

In order to determine more accurately the sizes of the colloidal complexes, we have

performed dynamical light scattering as function of the concentration. In Fig. 5 the

quadratic diffusion coefficient D(c) is shown versus c for three series of samples. These

solutions were obtained by dilution of the samples prepared at c = 1 wt. % (X = 0.2, empty

circles; X = 0.5, triangles) and at c = 4 wt. % (X = 0.25, closed circles). In this

concentration range, the diffusion coefficient varies according to :

D(c) = D0(1 + D2c) (3)

where D0 is the self-diffusion coefficient and D2 is a virial coefficient of the series

expansion [37]. From the sign of the virial coefficient, the type of interactions between the

aggregates, either repulsive or attractive can be deduced. Here D2 is small (of the order of

10-7 cm2s-2), indicating that the interactions between colloids are weak in this concentration

range. From the values of the self-diffusion coefficient D0, RH is derived and it is found in

the range 28 nm - 42 nm (see Table I for details). The data in Fig. 5 shows finally two

important properties for the nanoparticle-polymer complexes.

1 - Once the supermicellar aggregates are formed, they remain stable upon dilution

down to concentrations of the order of 0.01 wt. %. This result indicates that the critical

aggregation concentration (cac) for the self-assembled colloids is certainly below this

value. Working at concentrations lower than 0.01 wt. % in light scattering becomes

difficult because of the low level of the scattering signal.

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2 - The second important result found in Fig. 5 is that hydrodynamic sizes of the

complexes depend slightly on the mixing conditions. For instance, mixing at high

concentrations yields larger colloids. Although this observation is rather general (it was

also observed with surfactants [23]), it is not yet fully understood.

III.3 - Aggregation numbers

In this section, we determine the average aggregation numbers for the nanoparticle-

polymer complexes. We assume that each mixed solution at X > XC is a dispersion of

nanoparticle-polymer aggregates and that these aggregates are characterized by the

aggregation numbers

€

nnano and

€

npol . To this aim, we focus on the series of solutions that

have been prepared at the preferred composition XP, or close to it. Figs. 6 and 7 use the

Zimm representation [35] to display the light scattering intensities for these samples. The

solutions are those of Fig. 5 at X = 0.2 (Fig. 6) and X = 0.25 (Fig. 7). Here, the quantity

Kc/R(q,c) is plotted as function of q2 + cste×c and it is compared to the predictions of Eq.

1. These predictions, shown as straight lines in the two figures allow us to determine the

radius of gyration RG and the molecular weight

€

Mw,app for the aggregates. Their values are

reported in Table I. In both cases, the virial coefficient A2 is weak and of the order of 10-6

cm-3 g-2. In Table I, we also include the results for the X = 0.5-series. The radius of

gyration ranges between 20 and 30 nm, and the ratio RG/RH is found around 0.6 – 0.7. It is

interesting to note here that this later values are characteristic for core-shell structures.

They are typical for instance for polymeric micelles composed from block copolymers in

selective solvent [38-40].

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The apparent molecular weight of the complexes are of the order of 1 – 10×106 g mol-1.

For colloids resulting from a self-assembly process, the apparent molecular weight can be

expressed as [36]:

€

Mw,app = n2 n( )mn* + mw* −mn*( ) (4)

where

€

n and

€

n2 are the first and second moments of the distribution of aggregation

numbers. In Eq. 4,

€

mn* and

€

mw* are the number and weight-average molecular weights of

the elementary building blocks. Eq. 4 actually shows a classical result, which is that light

scattering can only provide a combination between some moments for the aggregation

number distribution, and not the moments themselves. For large values of

€

n , the second

term of the right hand side in Eq. 4 becomes negligible and the apparent molecular weight

can be rewritten :

€

Mw,app = n2 n( )mn* (5)

Eq. 5 is probably more appropriate than the one generally used in the literature , and which

expresses

€

Mw,app as the weighted sum of the weight-average molecular weights

[12,17,18]. By choosing for building block a unit composed by one nanoparticle and two

polymers (as we assume it is the case at X = XP), the aggregation number featuring in the

expression of

€

Mw,app (Eq. 5 and 6) coincides with

€

nnano. Using the values in Table I for

the single components and a polydispersity of 1.6, the number-average molecular weight

of the building blocks

€

mn* and the aggregation number

€

nnano2 nnano can be estimated. We

find that

€

nnano2 nnano is of the order of 30 – 60, whereas the number of polymers is about

twice this value, between 60 and 120 (see Table I). These data are important since they

provide a picture of the microstructure of the mixed aggregates. Note that under the

condition of electroneutrality for the mixed aggregates, the number of two polymers per

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nanoparticle in the complexes sets up the structural charge for the yttrium hydroxyacetate

about 140 (i.e. twice the degree of polymerization of the polyelectrolyte block). This

suggests that the structural charge of the yttrium hydroxyacetate nanoparticles might be

larger than for surfactant micelles. To have more accurate estimates of the core dimension

and distribution function, scattering experiments at higher q (> 10-3 Å-1) are necessary.

IV – Conclusions

In this paper we have shown the analogy between surfactant micelles and nanoparticles in

the process of complexation with polyelectrolyte/neutral block copolymers. To

demonstrate this analogy, we have used the same block copolymers for the two

approaches. The copolymer is poly(sodium acrylate)-b-poly(acrylamide) with molecular

weights 5000 and 60000 g mol-1 and it is obtained by controlled radical polymerization in

solution. The yttrium hydroxyacetate particles were chosen primarily as a model system

for inorganic colloids, and also because of potential applications in the fields of ceramics

and opto-electronic materials. Mixed aggregates are forming spontaneously by mixing

dilute solutions of polymers and of nanoparticles. Inspired by the results obtained with

micelles, the mixed colloids are also called “supermicellar” aggregates. In our previous

reports, we used the equivalent terminology of colloidal complexes. Light scattering

experiments is used quantitatively to determine the aggregation numbers of the

“supermicelles”. We have found that the aggregation numbers for the nanoparticles are in

the range 30 – 60, depending on the mixing conditions, whereas the numbers of polymers

is typically twice these values. As a comparison, in surfactant-polymer mixtures,

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aggregation numbers in the “supermicellar” aggregates were estimated of the order of

hundreds, with typically one surfactant micelle per polymer. This suggests that the

structural charge of the yttrium hydroxyacetate nanoparticles might be larger than for

micelles. In a recent paper, we have also shown that the nanoparticle-polymer

supermicelles exhibits a very goog long-term colloidal stability. Actually, it was found that

they were by far more stable than the nanoparticles themselves [10]. The reason of this

enhanced colloidal stability is the formation of a corona of the neutral blocks in the self-

assembly process. The overall aggregates are neutral, or weakly charged and they interact

with each others via soft steric interactions. The neutral corona has a typical thickness of

about 20 – 30 nm and it surrounds a core (of radius ~ 10 nm) containing the nanoparticles

and the polyelectrolyte blocks. We suggest that the results shown here are quite general

and that they could be easily extended to other types of nanoparticles, as for instance the

class of the lanthanide hydroxyacetates.

Acknowledgements : We thank Yoann Lalatonne, J. Oberdisse, R. Schweins, A. Sehgal

for many useful discussions, and Michel Rawiso for having pointed out to us the derivation

of the apparent molecular weight for polydisperse associative colloids. Mathias Destarac

from the Centre de Recherches d’Aubervilliers (Rhodia, France) is acknowledged for

providing us the polymers. This research is supported by Rhodia and by the Centre de la

Recherche Scientifique in France.

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Tables and Table Captions

dn/dc(cm3 g-1)

RH(nm)

RG(nm)

RG/RH

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Mw,app(g mol-1)

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nnano2 nnano

block copolymerPANa(69)-b-PAM(840)

0.159 8 n.d. n.d. 68×103 ..

nanoparticleY(OH)1.7(CH3COO)1.3

0.123 ~ 2 1.65 ± 1 ~ 0.8 27×103 ..

mixed aggregate X = 0.2

n.d. 33.5 ± 1.5 19.0 ± 1.5 0.57 3.2×106 31

mixed aggregate X = 0.25

0.150 42.0 ± 2 30.5 ± 1.5 0.73 5.8×106 57

mixed aggregate X = 0.5(*) n.d. 28.5 ± 1.5 20.0 ± 1 0.70 2.5×106 29

Table I : Characteristic sizes, molecular weights and aggregation numbers of mixed

aggregates made from nanoparticles and poly(acrylic acid)-b-poly(acrylamide) block

copolymers. The hydrodynamic (RH) and the gyration (RG) radii, as well as the apparent

molecular weight (

€

Mw,app ) of the colloidal complexes were determined using light

scattering. The aggregation numbers

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nnano2 nnano are derived according to Eq. 5. (*) : For

the series at X = 0.5, the aggregation number is calculated assuming that the mixed

aggregates are at the preferred composition XP = 0.2.

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Figure Captions

Figure 1 : Representation of a colloidal complex formed by association between

oppositely charged block copolymers and surfactants. The core is described as a complex

coacervation micro-phase of micelles connected by the polyelectrolyte blocks. The corona

is made of the neutral segments.

Figure 2 : Chemical structure of the poly(acrylic acid)-b-poly(acrylamide) diblock

copolymers investigated in this work. The degrees of polymerisation of each block are m =

69 and n = 840, yielding a structure that is described in the text as PANa(69)-b-PAM(840).

The polydispersity index for the diblock is 1.6.

Figure 3 : Evolution of (a) the Rayleigh ratio R(q,c) and (b) the hydrodynamic radius RH

as function of the charge ratio Z for mixed solutions of diblock copolymers PANa(69)-b-

PAM(840) and oppositely charged surfactant DTAB (T = 25° C). The Rayleigh ratio is

measured at q = 2.3×10-3 Å-1 (θ = 90°). Each data points in the figures corresponds to a

distinct solution, with total concentration c = 1 wt. %.

Figure 4 : Same as in Fig. 3 for solutions containing PANa(69)-b-PAM(840) copolymers

and yttrium hydroxyacetate nanoparticles (c = 1 wt. %). The parameter X is the mixing

ratio, i.e. the volume of yttrium-based sol relative to that of the polymer solution. The

concentrations in nanoparticles and polymers in the mixed solutions are cpol = c/(1+X) and

cnano = Xc/(1+X).

Figure 5 : Concentration dependence of the quadratic diffusion coefficient D(c) measured

for mixed yttrium-polymer solutions prepared at c = 1 wt. % (X = 0.2, empty circles; X =

0.5, triangles) and at c = 4 wt. % (X = 0.25, closed circles). The extrapolation at zero

concentration is the self-diffusion coefficient D0, from which the hydrodynamic radius RH

os derived. RH is found in the range 28 nm - 42 nm (see Table I for details).

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Figure 6 : Zimm plot showing the evolution of the light scattering intensity measured for

nanoparticle-polymer complexes as function of the wave-vector q (X = 0.2). Straight lines

are calculated according to Eq. 1 using for fitting parameters RG = 19 ± 1.5 nm,

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Mw,app =

3.2×106 g mol-1.

Figure 7 : Same as in Fig. 6 for a series of solutions prepared at c = 4 wt. % (X = 0.25)

and diluted thereafter down to c = 0.0165 wt. %. Straight lines are calculated according to

Eq. 1 using RG = 30.5 ± 15 Å,

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Mw,app = 5.8×106 g mol-1 for fitting parameters.

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Figure 1

Figure 2

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Figure 3a

Figure 3b

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24

Figure 4a

Figure 4b

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25

Figure 5

Figure 6

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Figure 7